This entry is from Summer semester 2018 and might be obsolete. No current equivalent could be found.

# Convex Optimization in Banach Spaces (dt. Konvexe Optimierung in Banachräumen)

 Level, degree of commitment Specialization module, compulsory elective module Forms of teaching and learning,workload Lecture (3 SWS), recitation class (1 SWS), 180 hours (60 h attendance, 120 h private study) Credit points,formal requirements 6 CP Course requirement(s): Written or oral examination Examination type: Successful completion of at least 50 percent of the points from the weekly exercises. Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics. Duration,frequency One semester, Im Wechsel mit anderen specialization moduleen zur Optimierung Person in charge of the module's outline Prof. Dr. Thomas Surowiec

## Contents

I. Infinite-Dimensional Optimization

• Semi-continuous functions in topological vector spaces
• Variational problems in (compact) metric and topological vector spaces
• Ekeland's Variational Principle
• First-order necessary and approximate conditions for variational problems
• Continuity of integral functionals on L-p spaces, Krasnoselskii's Theorem
• The role of the weak topology in existence theory

II. Convex analysis and Optimization

• Convex sets, convex functionals, Fenchel-Legendre conjugates (theorem of Fenchel-Moreau-Rockafellar, the Fenchel-Young-inequality)
• Generalized derivatives, e.g., directional differentiability, subdifferentials
• Calculation Rules for Convex Subdifferentials, Applications in Optimization

III. Numerical solution methods

• First-order numerical solution methods, e.g., projected (sub)gradients, mirror descent
• Second order numerical solution methods: Semismooth-Newton

## Qualification Goals

The students shall

• learn classical propositions of existence of the calculus of variations as well as some important concepts and results from nonlinear functional analysis, such as Nemytski operators and their role in the optimization and analysis of nonlinear partial differential equations,
• learn the extension of concepts from finite-dimensional convex analysis to infinite-dimensional problems; here a focus is placed on duality theory and subdifferentials,
• learn the formulation, implementation and convergence analysis of important algorithms in function spaces,
• Reassess knowledge from the basic modules and some advanced modules, e.g. from the modules for analysis and linear algebra as well as the optimization modules,
• learn the application of concepts from functional analysis, e.g. Dual Spaces, Hahn-Banach Set and Separation Sets,
• recognise relations with other areas of mathematics and other sciences,
• practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
• improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

## Prerequisites

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die entweder in den Basismodulen Lineare Algebra I, Lineare Algebra II, Analysis I und Analysis II oder Grundlagen der linearen Algebra, Grundlagen der Analysis und Grundlagen der Höheren Mathematik vermittelt werden. Außerdem werden die Kompetenzen aus dem Modul Maß- und Integrationstheorie empfohlen. Darüber hinaus sind Kenntnisse der Funktionalanalysis von Vorteil.

## Applicability

The module can be attended at FB12 in study program(s)

• M.Sc. Mathematics

When studying M.Sc. Mathematics, this module can be attended in the study area Specialization Modules in Mathematics.

The module can also be used in other study programs (export module).

Die Wahlmöglichkeit des Moduls ist dadurch beschränkt, dass es der Angewandten Mathematics zugeordnet ist.